EE283 Laboratory Exercise 4-Page 1
EE283 Laboratory Exercise #4 (Revised)
AC Components Objectives:
1. To understand the controls and operation of a signal generator and oscilloscope
2. To understand the phase relationship between Voltage and Current in AC components:
Resistors, Capacitors, and Inductors
3. To understand the relationship of resistance / reactance to frequency
Theory:
4.0.1. Resistor: Let the time varying Voltage across a resistor R, as shown in Figure 4.1(a), be
v(t) = VM cos wt (4.1)
Where w is the angular frequency (radians per second), 2pf, frequency f is the number of
cycles per second (units of Hertz), and t is time (in seconds).
From Ohm’s Law, the current i(t) = v(t)/R = VM cos wt / R (4.2)
Then, i(t) = IM cos wt where IM = VM / R (4.3)
Therefore, i(t) and v(t) are in phase, as shown in Figure 4.1(b).
In phasor form, I = V / R (4.4)
Phasors V and I are in phase, as shown in Figure 4.1(c). (Note V, I for phasors in bold.)
Figure 4.1 V-I Phase relations in a resistor
4.0.2. Capacitor: Let the time varying Voltage across a capacitor C, as shown in Figure 4.2(a), be
v(t) = VM cos wt (4.5)
Then, i(t) = C dv(t)/dt = -wC VM sin wt = wCVM cos (wt+90°) (4.6)
So i(t) = IM cos(wt+90°) where IM = wCVM = VM / XC (4.7)
EE283 Laboratory Exercise 4-Page 2
XC = 1/(wC) is called the capacitive reactance, and is inversely proportional to frequency.
Then i(t) LEADS v(t) by 90° as shown in Figure 4.2(b).
In Phasor form, V = I XC (4.8)
Where XC = -j /(wC) = 1/(jwC) (XC will have units of Ohms after unit cancellations.) (4.9)
Then, I leads V by 90°, as shown in Figure 4.2(c).
Figure 4.2 V-I Phase relations in a capacitor
4.0.3. Inductor: Let the time varying Voltage across an inductor L, as shown in Figure 4.3(a), be
v(t) = VM cos wt (4.10)
Then, i(t) = (1/L) ò v(t) dt = VM sin wt /(wL) = VM cos (wt - 90°) /(wL) (4.11)
So i(t) = IM cos(wt - 90°) where IM = VM/(wL) = VM / XL (4.12)
XL = wL is called the inductive reactance, and is directly proportional to frequency.
Current i(t) LAGS the Voltage v(t) by 90° as shown in Figure 4.3(b).
In Phasor form, V = I XL (4.13)
Where XL = jwL (XL will have units of Ohms after unit cancellations.) (4.14)
Then, I lags V by 90°, as shown in Figure 4.3(c).
Figure 4.3 V-I Phase relations in an inductor
EE283 Laboratory Exercise 4-Page 3
4.0.4. Root-mean-square (RMS) value
The root-mean-square value of an alternating Voltage or Current with period T is defined as:
XRMS=$%
𝑥' 𝑡 𝑑𝑡%* (4.15)
If x(t) = Xmax cos (wt + q), XRMS = 0.707 Xmax (applies to zero mean sinusoids) Note that in
equation 4.15, “X” stands for either Voltage or Current, not Admittance or Impedance.
The purpose of an “RMS” measurement is to give a Voltage directly comparable to DC
Voltage (or current) measurements. What AC Voltage dissipates the same power in a given
resistor as 120 Volts DC? Household power is “120 Volts,” but it is actually 170 VAC peak
and 340 VAC peak-to-peak. If you average the instantaneous power (V2/R) delivered to the
load over a full cycle (using integration) you get the RMS value of equation 4.15 above.
Procedure: 4.1 Basic set-up, oscilloscope, and function generator:
What we need to do is set up a loop circuit where we can observe and measure the AC
Voltage across and current through a given component (the “unit under test”) Figure 4.4
Illustrates what we need. (This will apply for all three devices.) The current is measured by
measuring the current across a small resistor and doing Ohm’s law: i(t) = v(t)/R.
Figure 4.4 Determination of Voltage and Current for a component being tested (ideal case)
There is a problem: the signal generator and the digital oscilloscope grounds will be
connected in common. As shown above, the Voltage measurement across the component has
neither probe in common with both a connection to the signal generator and the other
measurement channel. (The older analog oscilloscopes provided for “differential” channel
inputs, so that both terminals of a channel could “float”. The modern digital oscilloscopes do
not provide differential inputs (except using special, expensive probes).
unit under test10Ωresistor
i(t)=vR(t)/10Ω
vcomponent(t)measure
measurei(t)AC
ground (reference)
EE283 Laboratory Exercise 4-Page 4
What we will actually measure will be the Voltage of the unit under test, v1(t) =
vcomponent(t), and MINUS the resistor Voltage v2(t) = vR(t) = - icomponent(t). See Figure 4.5 below.
We can do this because the outer conductor of the signal generator is NOT actually grounded
through the instrument as earth ground, as are the outer conductors of the oscilloscope. So,
we can construct a circuit in which neither signal generator lead is grounded. Instead, we put
ground (the shielded lead of each oscilloscope channel) at the node in between the unit under
test and the resistor. The only disadvantage is that in the circuit current i(t), when positive
(in the direction of the arrow), will cause a negative Voltage to be indicated on Channel 2 of
the oscilloscope. On the Channel menu, an option to “invert” the channel fixes that. Note:
Do NOT use the T connector and BNC-BNC cable from the signal generator to Channel 1!
Figure 4.5 Test Circuit for Voltage and Current measurement
Refer to handouts on the oscilloscope and function generator. Make the following settings:
Set Oscilloscope to X versus T mode, time base to 1 msec per division, both channels to
DC coupling, and 5 Volts per division (to start with). On the channel menu for Channel 2,
make “invert” ON. Make sure both channels are set to “X1” for probes.
Set the signal generator, for Channel 1, to output a sine wave, at an amplitude of 10
Volts, no offset. Be sure to turn on the channel output.
4.2 Relation Between Voltage and Current for a Resistor
1. Measure the resistance of the 10 Ohm resistor used to measure current with the lab DMM.
Also, check the measured resistance of the resistor substitution box to confirm that it has the
desired resistance.
2. Connect the circuit shown in Figure 4.5. Choose a resistor R, about 470W (as directed), to
be the “unit under test.” Let channel 1 of the oscilloscope be the resistor under test Voltage.
That is done with a BNC cable from Oscilloscope Channel 1 directly (NOT connected to the
oscilloscope
signalgenerator
10Ω
unit under test
signal gen
Channel 2 (Resistor Voltageindicating - Current)
shield
shield
center
centeri(t)
i(t)=vR(t)/10Ω
shield
This node at bottom is NOT ground!
Channel 1 (indicating Voltage across unit under test)
GROUND!center
conductor
EE283 Laboratory Exercise 4-Page 5
“T” connector on the signal generator). Channel 1 of the oscilloscope needs to be across the
resistor under test with Ground toward the 10 Ohm resistor. Use Channel 2 to measure the
10 Ohm resistor Voltage. Put ground toward the unit under test, so that both oscilloscope
grounds are connected together. Warning: Be careful which lead is which. If you get the
shield and center conductor backwards, you will short out the channel! Connect the signal
generator with the center lead going to the resistor under test, and the outer lead (which, in
this circuit, is NOT grounded) going to the end of the 10 Ohm resistor away from the
oscilloscope grounds.
3. Set the function generator to configure Channel 1. (There is a button that selects which
channel you are setting.) Select a sine waveform. Set Amplitude to 10 Volts (that’s peak
Voltage. Peak to peak Voltage should be 20 Volts.) Set the frequency to the initial
frequency given for the lab exercise (about 1KHz). (The Offset should remain at 0 Volts.)
4. The default mode of the oscilloscope should be signal vs. time (YT). If that’s not its
mode, set “Display” Mode to YT. Set Channel 1 of the oscilloscope to 5 Volts per division.
Make sure it has “DC coupling” and “X1” probe setting. (Higher quality oscilloscope probes
are typically “X10”. Set the time base to 1 msec per division (of 1 cm). You should see the
sinusoid appear on the screen. If it is not a stable waveform, adjust the “trigger” Voltage
(knob). You might need to go to the Trigger menu to select Channel 1 as the trigger source.
(You may also need to push the Channel 1 button to get Channel 1 to appear on the screen.)
You should be able to observe that the signal displayed matches a 10 Volt peak (20 Volts p-
p) 1KHz (or other frequency) sinusoidal waveform. Now adjust the oscilloscope time base to
display just two or three cycles of the waveform.
5. Set channel 2 of the oscilloscope to 1 Volt per division. It should also be DC coupled,
with “X1” probe. On the channel menu, turn “invert” ON. You may need to push the
Channel 2 button for the trace to appear. You might be able to distinguish a trace, but the
amplitude will be low. Use the (vertical) position controls to make sure the two channels
don’t overlap. (You can turn down the scaling of Channel 1.) Now, you can adjust the
Channel 2 sensitivity until you see a significant trace, probably when you get to 10mV per
division. (You may see some noise on the signal.)
6. To review, Channel 1 is the tested resistor Voltage, and Channel 2 is the current (as
indicated by minus the Voltage across the resistor). Adjust the waveforms to get a pleasing
EE283 Laboratory Exercise 4-Page 6
display. You should see at least one complete cycle for each waveform. Position the
channels vertically so that “zero” (indicated by an arrow mark at left) is at one of the grid
lines (for reference). Now, sketch the Voltage (across the component) and Current
waveforms on your laboratory report. Annotate these sketches now (or later) to be properly
annotated graphs, meaning labels, units, numbers, grid lines and all the information needed to
fully understand what v(t) and i(t) are doing. These properly annotated graphs are NOT just
a copy of what you see on the screen. Your job is to interpret the information, not to be a
camera. For current, your annotated graph needs to show the value in terms of current
through the circuit, not Voltage across the resistor. Plot both with same zero point on the
vertical axis, as in Figure 4.6 later. (I will also accept one graph under the other with
separate x axes (for time) as long as the time axes are calibrated to the same scale to allow
time comparisons, to get phase, to be made.) See the discussion for finding current in
“Calculations” below. (Don’t sketch noise. Rather, estimate the centroid trace within the
noise.)
7. Now, change the oscilloscope mode to XY. This shows the current vs Voltage
relationship. Again, give a properly annotated graph for Current vs Voltage (Not Voltage vs
Voltage). This is called a Lissajous pattern. When doing this, you can set both channels to
zero (Ground) to properly position the origin (with both signals grounded, it’s just a dot on
the screen) at the intersection of the two center grid lines.
Calculations: Phase relationships:
The Voltage across a 10 Ohm resistor represents the current through the circuit, and hence
through the unit under test. From the waveforms recorded, find the phase relationship. One
full cycle represents 360°. An example is shown in Figure 4.6, where 16 divisions
correspond to 360°, and the phase difference is 45° (2 divisions). I(t) lags v(t) by 45°.
Figure 4.6 Example to illustrate the calculation of phase difference
EE283 Laboratory Exercise 4-Page 7
Phase difference can also be calculated using the Lissajous pattern, as shown in Figure 4.7.
Figure 4.7 Lissajous pattern
If q is the phase angle, then sin q = Y1/Y2.
For the resistor, find the phase angle by both time domain traces and from the Lissajous
pattern, and put those into your report. You should verify that, for a resistor, the current is in
phase with the Voltage. With the Lissajous pattern displayed, change the frequency up and
down an order of magnitude or so. Did the Lissajous figure change? If not, the current
versus Voltage for a resistor is frequency invariant. (We will look at that again in Part 4.5.)
4.3 Relation Between Voltage and Current for a Capacitor Replace the resistor (the unit under test, not the 10 Ohm current sensing resistor) with a 0.4
microFarad (or some other value as assigned) capacitor. Repeat steps 6 and 7, making any
adjustments necessary, to generate traces for current and Voltage versus time for the
capacitor, and then (similarly) a Lissajous pattern. Draw the waveforms on your report,
again, making sure they are properly annotated. Be sure to indicate whether current LEADS
or LAGS the Voltage. Calculate the phase relationship between current and Voltage both
ways. After doing so, change the frequency up and down an order of magnitude, and notice
how the Lissajous pattern changes. Do the changes make sense? What is changing, and
why? (You can look at this as you change frequencies in the section below.)
4.4 Impedance vs Frequency Relationship for a Capacitor Now, vary the frequency of the signal generator over a series of values as prescribed
(between 250 Hz and 4KHz) and record the Voltage and the current measuring resistor
Voltage as indicated on the oscilloscope at each frequency. Record these in a table (first
three columns) using the peak-to-peak Voltages recorded from the oscilloscope. Later,
convert Voltage to RMS, and calculate current as and RMS current. See “Calculations”
below. The ratio of Voltage to Current (both RMS) is the “Impedance” of the capacitor.
EE283 Laboratory Exercise 4-Page 8
(This is actually the absolute value of, or magnitude of, Impedance. The calculation can also
be made with peak to peak or peak Voltages and currents, but both Voltage and current must
be consistent.) The “Reactance” is the theoretical relationship between the Voltage and
Current, which can be calculated given the capacitor value C and the frequency. The
Impedance found from lab results should be very close to the Reactance found from the
capacitor value and frequency.
4.5 Relation Between Voltage and Current in an Inductor Replace the capacitor with an inductor with a specified inductance (about 70mH or so).
Repeat steps 6 and 7 taken for the resistor, making any adjustments necessary to generate
traces for current and Voltage versus time for the inductor. Then (similarly) produce a
Lissajous pattern. Draw the waveforms on your report, again, making sure they are properly
annotated graphs. Calculate the phase relationship between current and Voltage both ways.
Be sure to indicate whether current LEADS or LAGS the Voltage. After doing so, change
the frequency up and down, and notice how the Lissajous pattern changes. Do the changes
make sense? What is changing, and why? (Do that as part of the following section.)
4.6 Impedance vs Frequency Relationship for an Inductor
Now, vary the frequency of the signal generator over a series of values as prescribed
(between 250 Hz and 4KHz) and record the Voltage and the current measuring resistor
Voltage as indicated on the oscilloscope at each frequency. Record frequency and these
Voltages in a table using the peak-to-peak Voltages recorded from the oscilloscope (the first
three columns). Later, convert Voltage to RMS, and calculate current as RMS current. See
“Calculations” below. The ratio of Voltage to Current (both RMS) is the “Impedance” of the
inductor. (It is actually the absolute value of, or magnitude of, Impedance. The calculation
can also be made with peak to peak or peak Voltages and currents, but both Voltage and
current must be consistent.) The “Reactance” is the theoretical relationship between the
Voltage and Current, which can be calculated given the inductor value L and the frequency.
The Impedance found from lab results should be very close to the Reactance found from the
inductor value and frequency.
EE283 Laboratory Exercise 4-Page 9
Calculations (for 4.4, 4.6):
After recording Vp-p across the resistor (used to measure current) and the Voltage (also p-
p) across the unit under test, for each frequency, complete the tables by calculating the RMS
Voltage and Current, from those values the impedance at each frequency. As seen earlier,
because the current measuring resistor is 10 Ohms, i(t) = v(t) / (10 Volts / Ampere). RMS
values for Voltage and Current can be calculated from peak to peak values as shown in
equation 4.15 earlier. In the last column of the table, calculate the Reactance for the unit
under test, given the nominal value of the component and the frequency. These numbers
should approximately match the results derived from your lab measurements. Do this for the
capacitor and the inductor. (We are skipping the resistor since the reactance of a resistor
does not change with frequency.)
The form report containing these tables and previous graphs and observations is due
at the beginning of the next laboratory session.
4.7 Impedance vs Frequency graphs:
Using the tables of lab data for measured Impedance, plot on one graph the curves for the
Capacitor and Inductor impedances versus frequency. Use a logarithmic scale for frequency
and impedance. Also try plotting this data on reactance graph paper if available. Both graph
data series are to be plotted against the same vertical and horizontal axes. This is to be done
using the Excel program. Refer to the Engineering Laboratory Reports Manual for
recommended techniques for plotting good graphs. The graphs are to be monochrome (black
and white).
The graph, including an identification of the students, their lab section and station
number, is to include nicely formatted tables (used to construct the graph) and the required
graph. It should all be on one page. This is due at the beginning of the next lab session.
Comments: This laboratory exercise has been revised to use the digital oscilloscopes rather than the
analog oscilloscopes used earlier. The problem with not having differential inputs was
discussed. The resistor frequency characteristics part of the earlier version of this exercise
has been omitted.
EE283 Laboratory Exercise 4-Page 10
EE283 Laboratory Exercise #4 AC Components Form report
Students: Section Date: Station
4.2 Relationship between Voltage and current for a Resistor R value =
10W resistor value:
(a) Voltage and current waveforms (b) Lissajous pattern
Phase from waveforms (show work): Phase from Lissajous pattern (show work):
Any Observations and Comments (concerning the whole report):
EE283 Laboratory Exercise 4-Page 11
4.3 Relationship between Voltage and current for a Capacitor Capacitor value:
(a) Voltage and current waveforms (b) Lissajous pattern
Phase from waveforms (show work): Phase from Lissajous pattern (show work):
4.4 Impedance vs Frequency relationships for capacitor
Table 4.4 Relationship between capacitance and frequency. C = µF
Frequency(Hz)
Voltage acrossRs: VRsVolts (p-p)
Current throughC (I = VRs / Rs)mA (RMS)
Voltageacross C:VCVolts(p-p)
Voltageacross C:VCVolts(RMS)
Impedance|Z| = VC / I
kΩ
Reactance|XC|=1/ωC
kΩ
EE283 Laboratory Exercise 4-Page 12
4.5 Relationship between Voltage and current in an Inductor Inductor value:
(a) Voltage and current waveforms (b) Lissajous pattern
Phase from waveforms (show work): Phase from Lissajous pattern (show work):
4.6 Impedance vs Frequency relationship for Inductor
Table 4.6 Relationship between inductance and frequency. L = mH
Frequency(Hz)
Voltage acrossRs: VRsVolts (p-p)
Current throughL (I = VRs / Rs)mA (RMS)
Voltageacross L:VLVolts(p-p)
Voltageacross L:VLVolts(RMS)
Impedance|Z| = VL / I
kΩ
Reactance|XL|=ωLkΩ